Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

Abstract

We introduce the notion of (ψ, Β)-derivative of a function of one complex variable, and define on the basis of this the classes$L_\beta ^{\psi \mathfrak{N}} (G)$ of (ψ, Β)-differentiable analytic functions in a bounded domain G. The classes$L_\beta ^{\psi \mathfrak{N}} (G)$ consist of the Cauchy-type integrals whose densities f(ζ) are such that the induced functions$\tilde f (t)$ on the unit circle are periodic functions of classes$L_\beta ^\psi \mathfrak{N}$. We consider approximation of functions$f \in L_\beta ^\psi \mathfrak{N} (G)$ by algebraic polynomials constructed from their series expansions in Faber polynomials.